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Classifying modules over \(K\)-theory spectra. (English) Zbl 0911.55005
This paper works in the category of \(S\)-algebras, the brave new world developed recently by May and others, which enables the algebraic topology of spectra to be done at the point-set level. One of the main general results states that if \(R\) is an \(S\)-algebra, then an \(R_*\)-module of projective dimension at most 2 can be realized as the module of homotopy groups of some \(R\)-module, uniquely if this projective dimension is less than 2. This applies to the \(crt\)-category (with objects \(ko\), \(ku\), and \(kt\)), for it is shown that all \(crt\)-acyclic \(crt\)-modules have projective dimension at most 2. For a fixed \(crt\)-module \(M\) of projective dimension exactly 2, there is an equivalence relation finer than homotopy equivalence so that equivalence classes of \(ko\)-modules \(X\) with \(\pi_*^{crt}(X)=M\) correspond to elements of Ext\(_{crt}^{2,-1}(M,M)\). One consequence is that any \(ko_*\)-module \(M_*\) can be realized as the homotopy groups of some \(ko\)-module.

55P42 Stable homotopy theory, spectra
55N15 Topological \(K\)-theory
19L41 Connective \(K\)-theory, cobordism
Full Text: DOI
[1] Adams, J.F., Stable homotopy and generalized homology, (1974), University of Chicago Press New York · Zbl 0309.55016
[2] Anderson, D., ()
[3] Bass, H., Big projective modules are free, Illinois J. math., 7, 24-31, (1963) · Zbl 0115.26003
[4] Blackadar, B., Matricial and ultramatricial topology, (1991), University of Nevada Berkeley, Preprint · Zbl 0803.46075
[5] Boardman, J.M., Conditionally convergent spectral sequences, (1981), Johns Hopkins University, Preprint · Zbl 0947.55020
[6] Bousfield, A.K., The localization of spectra with respect to homology, Topology, 18, 257-281, (1979) · Zbl 0417.55007
[7] Bousfield, A.K., On the homotopy theory of K-local spectra at an odd prime, Amer. J. math., 107, 895-932, (1985) · Zbl 0585.55004
[8] Bousfield, A.K., A classification of K*-local spectra, J. pure appl. algebra., 66, 121-163, (1990) · Zbl 0713.55007
[9] Bousfield, A.K., On K*-local stable homotopy theory, (), 23-33, Cambridge · Zbl 0754.55006
[10] Cartan, H.; Eilenberg, S., Homological algebra, (1956), Princeton University Press · Zbl 0075.24305
[11] Costenoble, S.; Waner, S., Generalized Toda brackets and equivariant Moore spectra, Trans. amer. math. soc., 333, 849-860, (1992) · Zbl 0767.55006
[12] Dădărlat, M.; Némethi, A., Shape theory and (connective) K-theory, J. operator theory, 23, 207-291, (1990) · Zbl 0755.46036
[13] Elmendorf, A.; Kriz, I.; Mandell, M.; May, J.P., Commutative algebra in stable homotopy theory, (1995), Preprint · Zbl 0865.55007
[14] Hilton, P.; Stammbach, U., A course in homological algebra, (1971), Springer Princeton, NJ · Zbl 0238.18006
[15] Lin, T.-Y., Adams type spectral sequences and stable homotopy modules, Indiana math. J., 25, 135-158, (1976) · Zbl 0333.55013
[16] Matlis, E., Injective modules over Noetherian rings, Pacific J. math., 8, 511-528, (1958) · Zbl 0084.26601
[17] Matsumura, H., Commutative ring theory, () · Zbl 0211.06501
[18] Matsumura, H., Graded rings and modules, (), 193-203 · Zbl 0803.13001
[19] May, J.P., E∞ ring spaces and E∞ ring spectra, (), with contributions from F. Quinn, N. Ray and J. Tornehave · Zbl 1228.55010
[20] Mitchell, B., Rings with several objects, Adv. in math., 8, 1-161, (1972) · Zbl 0232.18009
[21] Năstăsescu, C.; van Oystaeyen, F., Graded ring theory, (1982), North-Holland Berlin · Zbl 0494.16001
[22] Ravenel, D.C., Localization with respect to certain periodic homology theories, Amer. J. math., 106, 351-414, (1984) · Zbl 0586.55003
[23] Segal, G., K-homology theory and algebraic K-theory, (), 113-127
[24] Stolz, S., Splitting certain mspin-module spectra, Topology, 33, 159-180, (1994) · Zbl 0807.55005
[25] Wolbert, J., Classifying modules over E∞ ring spectra, (1994), Preprint
[26] Wolbert, J., ()
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